Adaptive Nonintrusive Reconstruction of Solutions to High-Dimensional Parametric PDEs

DOI10.1137/21m1461988zbMath1530.65014arXiv2112.01285OpenAlexW3215188971MaRDI QIDQ6039259

Sebastian Heidenreich, Martin Eigel, Nando Farchmin, Philipp Trunschke

Publication date: 4 May 2023

Published in: SIAM Journal on Scientific Computing (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/2112.01285

zbMATH Keywords

adaptivityparametric PDEsuncertainty quantificationtensor trainresidual error estimatorstochastic Galerkin finite element methodlow-rank tensor regression

Mathematics Subject Classification ID

Spectral, collocation and related methods for boundary value problems involving PDEs (65N35) Error bounds for boundary value problems involving PDEs (65N15) Numerical solutions to stochastic differential and integral equations (65C30) Complexity and performance of numerical algorithms (65Y20) Multilinear algebra, tensor calculus (15A69)

Cites Work

• Unnamed Item
• Tensor-Train Decomposition
• TT-cross approximation for multidimensional arrays
• Efficient low-rank approximation of the stochastic Galerkin matrix in tensor formats
• Adaptive stochastic Galerkin FEM
• Tensor networks and hierarchical tensors for the solution of high-dimensional partial differential equations
• Convergence rates of best \(N\)-term Galerkin approximations for a class of elliptic SPDEs
• Adaptive stochastic Galerkin FEM for lognormal coefficients in hierarchical tensor representations
• Low-rank tensor reconstruction of concentrated densities with application to Bayesian inversion
• Approximation and sampling of multivariate probability distributions in the tensor train decomposition
• Variational Monte Carlo -- bridging concepts of machine learning and high-dimensional partial differential equations
• Non-intrusive tensor reconstruction for high-dimensional random PDEs
• Adaptive stochastic Galerkin FEM with hierarchical tensor representations
• On manifolds of tensors of fixed TT-rank
• Goal-oriented adaptive finite element multilevel Monte Carlo with convergence rates
• A Review of Unified A Posteriori Finite Element Error Control
• Energy Norm A Posteriori Error Estimation for Parametric Operator Equations
• Sparse polynomial approximation of parametric elliptic PDEs. Part II: lognormal coefficients
• Approximating Infinity-Dimensional Stochastic Darcy's Equations without Uniform Ellipticity
• ANALYTIC REGULARITY AND POLYNOMIAL APPROXIMATION OF PARAMETRIC AND STOCHASTIC ELLIPTIC PDE'S
• Sparse tensor discretizations of high-dimensional parametric and stochastic PDEs
• Breaking the Curse of Dimensionality, Or How to Use SVD in Many Dimensions
• An Adaptive Multilevel Monte Carlo Method with Stochastic Bounds for Quantities of Interest with Uncertain Data
• A convergent adaptive stochastic Galerkin finite element method with quasi-optimal spatial meshes
• Error Decomposition and Adaptivity for Response Surface Approximations from PDEs with Parametric Uncertainty
• Polynomial Chaos Expansion of Random Coefficients and the Solution of Stochastic Partial Differential Equations in the Tensor Train Format
• Spectral Methods for Uncertainty Quantification
• A Sparse Grid Stochastic Collocation Method for Partial Differential Equations with Random Input Data
• An Anisotropic Sparse Grid Stochastic Collocation Method for Partial Differential Equations with Random Input Data
• Sampling-free Bayesian inversion with adaptive hierarchical tensor representations
• Efficient Adaptive Algorithms for Elliptic PDEs with Random Data
• Optimal weighted least-squares methods
• Convergence bounds for empirical nonlinear least-squares
• An adaptive stochastic Galerkin method based on multilevel expansions of random fields: Convergence and optimality
• Convergence and rate optimality of adaptive multilevel stochastic Galerkin FEM
• An Adaptive Stochastic Galerkin Tensor Train Discretization for Randomly Perturbed Domains
• Two-Level a Posteriori Error Estimation for Adaptive Multilevel Stochastic Galerkin Finite Element Method
• A Hybrid Alternating Least Squares--TT-Cross Algorithm for Parametric PDEs
• Efficient Adaptive Multilevel Stochastic Galerkin Approximation Using Implicit A Posteriori Error Estimation
• Convergence of Adaptive Stochastic Galerkin FEM
• Approximation of high-dimensional parametric PDEs
• Stochastic Collocation for Elliptic PDEs with Random Data: The Lognormal Case
• Tensor Spaces and Hierarchical Tensor Representations
• STOCHASTIC GALERKIN DISCRETIZATION OF THE LOG-NORMAL ISOTROPIC DIFFUSION PROBLEM
• On the convergence of the stochastic Galerkin method for random elliptic partial differential equations
• N-TERM WIENER CHAOS APPROXIMATION RATES FOR ELLIPTIC PDEs WITH LOGNORMAL GAUSSIAN RANDOM INPUTS
• The completion of locally refined simplicial partitions created by bisection
• Efficient Adaptive Stochastic Galerkin Methods for Parametric Operator Equations

This page was built for publication: Adaptive Nonintrusive Reconstruction of Solutions to High-Dimensional Parametric PDEs